Why do I keep getting these from people I haven't even met? Is the Dragonborn somehow related to EVERYONE in Skyrim, so closely that s/he gets noted in people's wills, regardless of race? I do like the extra gold, but it seems strange to hear the courier say "Sorry for your loss", when I have never met the recently deceased.

Earliest this has happened for me was when I was heading to Whiterun after escaping Helgen...

Getting people's trust is really easy sometimes, you just need to complete one of their quests, deliver something for(or to) them, or even just sell them some lumber, if someone gave you a letter like that on your way to Whiterun, it was probably someone you met in Riverwood or something, you'll never get letters of inheritance for nothing

Oh, ok. Thanks be to ye for clearing it up. So, I've done some favors for people and get a few Septims after they pass away... That's neat. Hmmm... Would it work if there was a sudden... wave of death amongst the people I've helped? Say, by accidental stabbing by someone who might be a Dragonborn? If someone I helped tripped and fell on my sword?

Nope, once you attack them they'll stop being your friends, so, even if you kill them, you won't get anything, but they do die a lot when a dragon attacks, or when vampires invade a city in one of the random encounters added by Dawnguard.

I forgot to mention that other ways to become friends with people are as easy as selling them crops, investing in their stores, and even giving them money if they are beggars.

Though, personally, I really don't think 90 gold (as the Jarl takes 10 for taxes) is worth having an NPC killed, I can probably make a lot more and faster by raiding bandits and selling their stuff

Actully I had Vilkas and Farkas follow me around LONG after finishing the companions quests. I had to kill them ( really didnt have a choice) and a few mins later got a inheritence letter from both of them.

After that I got one from everyone I met in the next few days. I thought I was cursed